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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. (English) Zbl 0779.65013
The Kummer function $$\Phi(a,c;z)$$, expressed in its well-known integral form, is numerically evaluated for complex values of $$z$$ by means of the trapezoidal rule. The achieved high degree of accuracy is explained through a detailed investigation of the related Euler-Maclaurin formula. Theoretically interesting error bounds are given. Recurrence relations are used for obtaining $$a$$ and $$c$$ values for which the trapezoidal rule is not suitable. The procedure is compared with one based on Gauss-Jacobi quadrature.
##### MSC:
 65D20 Computation of special functions and constants, construction of tables 33C15 Confluent hypergeometric functions, Whittaker functions, $${}_1F_1$$